In wavelength division multiplexed (WDM) optical communication systems, a number of different optical carrier wavelengths are separately modulated with data to produce modulated optical signals. The modulated optical signals are combined into an aggregate signal and transmitted over an optical transmission path to a receiver. The receiver detects and demodulates the data.
One data modulation scheme used in optical communication systems is phase-shift keying (PSK). In a PSK modulation scheme, the phase of an optical carrier is modulated such that the phase or phase transition of the optical carrier represents symbols encoding one or more bits. A variety of PSK schemes are well-known. In a binary phase-shift keying (BPSK) modulation scheme, for example, two phases may be used to represent 1 bit per symbol. In a quadrature phase-shift keying (QPSK) modulation scheme, four phases may be used to encode 2 bits per symbol. Other phase shift keying formats include differential phase shift keying (DPSK) formats and variations of phase shift keying and differential phase shift keying formats, such as return-to-zero DPSK (RZ-DPSK).
Polarization multiplexing (POLMUX) may be implemented with a PSK modulation format to double the spectral efficiency of the format. In a POLMUX format, two relatively orthogonal states of polarization of the optical carrier are separately modulated with data, e.g. using a PSK modulation format, and then combined for transmission. For example, in a POLMUX-QPSK signal, orthogonal polarizations of the same optical carrier are modulated with different data streams using a QPSK modulation format.
During transmission of the modulated signals over the transmission path, non-linearities in the path may introduce transmission impairments, such as chromatic dispersion (CD), polarization mode dispersion (PMD) and polarization dependent loss (PDL), into the signals. In general, chromatic dispersion is a differential delay in propagation of different wavelengths through the transmission path, and polarization mode dispersion is a differential delay in propagation of different polarizations through the transmission path. Polarization dependent loss is a differential attenuation for different polarizations through an optical path.
At the receiver, coherent detection with polarization diversity may be used to detect POLMUX-PSK modulated optical signals. In the coherent receiver, the incoming signal from the transmission path has an arbitrary state of polarization (SOP) that is separated into linear x- and y-polarization components with a polarization beam splitter (PBS) and each of the components is mixed with a linearly polarized local oscillator (LO) to measure the complex amplitudes of the x- and y-components. However, in a POLMUX system, each of the outputs of the polarization-diversity coherent receiver includes both of the polarizations on which data is modulated, i.e. the electrical output of the receiver is still polarization multiplexed.
Digital signal processing (DSP) may be applied to the outputs of the coherent receiver to de-multiplex the POLMUX signals, compensate for transmission impairments such as PMD, PDL and other residual impairments, and demodulate the data. It is known to perform polarization de-multiplexing in DSP using a two-by-two matrix and a constant modulus algorithm (CMA). In general, the CMA uses the assumption that the signal has a constant modulus (i.e. amplitude for a PSK signal) to filter the digitized versions of signals using the input to the CMA and feedback from the output of the CMA to provide outputs representing the data modulated on each of the respective data modulated polarizations of the transmitted signals.
One known disadvantage of using a CMA in this application is that the outputs of the CMA may converge together, even though the inputs to the CMA are different. This convergence is sometimes referred to as the “singularity problem.” A discussion of CMA convergence is discussed, for example, in Performance Analysis of Polarization Demultiplexing Based on Constant-Modulus Algorithm in Digital Coherent Receivers by Kikuchi, published in Optics Express, vol. 19, No. 10, pp 9868-9880, May 9, 2011, which indicates that convergence can be avoided by applying unitary constraint in the CMA. Unfortunately, applying a unitary constraint in the CMA results in a performance penalty, especially with the presence of PDL and high order PMD.